A combinatorial characterization of geometric spreads

Abstract A t-spread in a projective space P = PG(d, q) is a set of t-dimensional subspaces which partitions the point set of P. A t-spread S is called geometric if it induces a spread in any (2t+1)-dimensional subspace containing at least two elements of S. In this note we characterize the geometric t-spreads S among all partial spreads in the first nontrivial case PG(3t+2, q) by the property that any subspace of dimension 3t contains at least one element of S. This is the first instance of a combinatorial characterization of geometric spreads.